We find necessary and sufficient conditions for a

generalised Mehler semigroup to be covariant under the action of a locally compact group. These are then applied to implement "noise reduction" for Hilbert-space valued Ornstein - Uhlenbeck processes driven by Levy processes

Applebaum, D.

English

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promoting access to White Rose research papers    White Rose Research Online eprints@whiterose.ac.uk    Universities of Leeds, Sheffield and York http://eprints.whiterose.ac.uk/    This is an author produced version of a paper published in Markov Processes and Related Fields.    White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/9799     Published paper Applebaum, D. (2007) Covariant Mehler semigroups in Hilbert space, Markov Processes and Related Fields, 13 (1), pp. 159-169  Covariant Mehler Semigroups in Hilbert SpaceDavid ApplebaumProbability and Statistics Department,University of Sheffield,Hicks Building, Hounsfield Road,Sheffield, England, S3 7RHe-mail: D.Applebaum@sheffield.ac.ukDedicated to the memory of J.T.LewisAbstractWe find necessary and sufficient conditions for a generalised Mehlersemigroup to be covariant under the action of a locally compact group.These are then applied to implement “noise reduction” for Hilbert-space valued Ornstein-Uhlenbeck processes driven by Lévy processes.1 IntroductionGeneralised Mehler semigroups are beautiful objects which have attracted theattention of both analysts and probabilists. They are semigroups of linearoperators (T (t), t ≥ 0) acting on the space of bounded continuous functionson a real separable Hilbert space which are built from two components:• a C0-semigroup (S(t), t ≥ 0) acting in H with generator J ,• a family (µt, t ≥ 0) of probability measures on H satisfying the skew-convolution property µt+r = µt ∗ (µr ◦ (S(t)−1),through the formula(T (t)f)(x) =∫Hf(S(t)x + y)µt(dy).From a probabilistic point of view, they arise as the transition semigroupsof H-valued Ornstein-Uhlenbeck processes (Y (t), t ≥ 0) which are driven by1H-valued Lévy processes (Z(t), t ≥ 0) (i.e. processes with stationary andindependent increments) through the stochastic differential equation:dY (t) = JY (t)dt + dZ(t).When H is infinite-dimensional, such equations have been extensively studiedin the Gaussian case where Z is a standard Brownian motion (see e.g. [6]),and have been extended to the more general Lévy case in [5] and [2]. Inthe “classical case” where J is a negative scalar and H is finite dimensional,the use of Lévy noise has attracted recent attention through applicationsto volatility models in option pricing [3], [16] and branching processes withimmigration [15].From the analytic viewpoint, extensive work on Mehler semigroups hasbeen carried out by M.Röckner and his collaborators ([4], [9], [12], [13],[18]). In particular, they have shown that when invariant measures exist,the generators can be constructed as pseudo-differential operators acting ina suitable Lp space [12], while in [18], the strong Feller property is provedand the Harnack, Poincaré and logarithmic Sobolev inequalities established.Applications of generalized Mehler semigroups to measure-valued catalyticbranching processes have been developed in [7].From a modelling viewpoint, a Mehler semigroup (or Ornstein-Uhlenbeckprocess) describes the interaction of a system, which may itself be highly com-plex, with a noisy environment. It is natural to seek to simplify the problemby exploiting symmetry, when this is present. To this end, in this paper weassume that the Hilbert space H carries a unitary representation of a groupG and we ask for conditions under which the Mehler semigroup commuteswith this group action. Our main result is that necessary and sufficient con-ditions for this are that the semigroup (S(t), t ≥ 0) itself commutes with thegroup action and the probabilities (µt, t ≥ 0) are left invariant. When G actsirreducibly, this forces S(t) to be trivial, so we have a “classical” Ornstein-Uhlenbeck process and the driving noise is simplified. In the non-compactcase, we have extensive “noise reduction” in that the only admissible (i.e.suitably invariant) driving Lévy processes are those of pure jump type.Acknowledgement. I am grateful to Robin Hudson for a helpful obser-vation and to the referee for useful commentsNotation R+ = [0,∞). If T is a topological space, then B(T ) denotesits Borel σ-algebra. If H is a real separable Hilbert space, Bb(H) is the spaceof bounded Borel measurable real-valued functions on H and L(H) is the∗-algebra of all bounded linear operators on H. If M is a non-empty subsetof L(H), then M ′′ is the smallest von Neumann algebra containing M . I isthe identity operator in L(H). The domain of a linear operator T acting in2H is denoted as Dom(T ). A mapping from R+ to H is càdlàg if it is rightcontinuous and has a left limit at every point.If x ∈ H, δx denotes Dirac mass at x which is the probability measure onB(H) defined by δx(A) = 1 if x ∈ A, and δx(A) = 0 if x /∈ A. µ1 ∗µ2 denotesthe convolution of Borel measures µ1 and µ2 defined on H.If (Ω,F , P ) is a probability space and X : Ω → H is a random variable,the law of X is the probability measure pX defined on B(H) by pX(A) =P (X ∈ A). If X and Y are two such random variables, we write X d= Y ifpX = pY .2 Generalised Mehler Semigroups ([4], [9])Let H be a real separable Hilbert space and (S(t), t ≥ 0) be a C0-semigroupacting in H with infinitesimal generator J . Let (µt, t ≥ 0)) be a family ofprobability measures defined on B(H). Consider the space Cb(H) of boundedcontinous real valued functions on H. It is a Banach space when equippedwith the supremum norm. We define a family of bounded linear operators(T (t), t ≥ 0) on Cb(H) by the prescription(T (t)f)(x) =∫Hf(S(t)x + y)µt(dy), (2.1)for each t ≥ 0, f ∈ Cb(H), x ∈ H. It is shown in [4] that (T (t), t ≥ 0) is asemigroup if and only if (µt, t ≥ 0) is a skew-convolution semigroup in thesense thatµt+r = µt ∗ (µr ◦ (S(t)−1) (2.2)for all t, r ≥ 0.If ν is a finite Borel measure on H we denote its Fourier transform byν̂ : H → C, so thatν̂(y) =∫Hei〈x,y〉ν(dx),for each y ∈ H. In [4] it is further established that if for all y ∈ H,• the mapping t → µ̂t(y) is locally absolutely continuous on [0,∞) anddifferentiable at t = 0,• the mapping t → λ(S(t)∗y) is locally Lebesgue integrable on [0,∞),whereλ(y) := − ddtµ̂t(y)∣∣∣∣t=0,3then (2.2) is equivalent toµ̂t(v) = exp{−∫ t0λ(S(r)∗v)dr}, (2.3)for all v ∈ H. The mapping λ : H → C is negative definite, hermitian andsatisfies λ(0) = 0.In the sequel we will always assume that the conditions given above hold.(T (t), t ≥ 0) is then called a (generalised) Mehler semigroup. We also assumethat λ is Sazonov continuous, i.e. it is continuous with respect to the locallyconvex topology on H generated by the seminorms y → ||Ay||, where Aruns over all Hilbert-Schmidt operators in H. We can now utilise the Lévy-Khinchine formula in H (see e.g. Chapter 6 in [17]) to assert that λ musttake the formλ(y) = −i〈b, y〉+ 12〈y, Qy〉+∫H−{0}(1− ei〈u,y〉 + i〈u, y〉1{||u||<1})ν(du), (2.4)where b ∈ H,Q is a positive, self-adjoint trace class operator on H and νis a Lévy measure on H − {0}, i.e. ∫H−{0}(||x||2 ∧ 1)ν(dx) < ∞. The triple(b,Q, ν), which is the (set of) characteristics of λ, determines λ uniquely.Note that (T (t), t ≥ 0) is not strongly continuous on Cb(H) with the usualuniform topology. Provided (µt, t ≥ 0) is weakly convergent to Dirac massat the origin, it can be shown to be continuous with respect to the so-calledmixed topology (see [10], [11]). We will not need such results in this paper.3 Covariant Mehler SemigroupsLet G be a locally compact group and U be a continuous unitary (non-trivial)representation of G in H. For each g ∈ G, define Ug : Cb(H) → Cb(H) by(Ugf)(x) = f(Ugx),for each f ∈ Cb(H), x ∈ H, then g → Ug is a anti-homomorphism of Ginto the automorphism group of the Banach algebra Cb(H). We say thatthe Mehler semigroup (Tt, t ≥ 0) is covariant under the action U of G (orG-covariant, for short) ifT (t)Ug = UgT (t),for each t ≥ 0, g ∈ G. In the sequel, ρg := ρ ◦ U−1g , for all g ∈ G, whenever ρis a function or measure defined on H.4Theorem 3.1 The following are equivalent:(i) The Mehler semigroup (T (t), t ≥ 0) is G-covariant,(ii) µt ∗ δS(t)Ugx = δUgS(t)x ∗ µgt ,(iii) ei〈v,S(t)Ugx〉µ̂t(v) = ei〈v,UgS(t)x〉µ̂gt (v),for all g ∈ G, x, v ∈ H, t ≥ 0.Proof. (i) ⇒ (ii). For all f ∈ Cb(H), x ∈ H,(UgT (t)f)(x) =∫Hf(S(t)Ugx + y)µt(dy) =∫Hf(y)(µt ∗ δS(t)Ugx)(dy).(T (t)Ugf)(x) =∫Hf(UgS(t)x + Ugy)µt(dy)=∫Hf(UgS(t)x + y)µgt (dy)=∫hf(y)(δUgS(t)x ∗ µgt )(dy).(ii) ⇒ (i) is immediate.(ii) ⇒ (iii) Put f(·) = ei〈v,·〉.(iii) ⇒ (ii) follows from the fact that a finite measure is uniquely deter-mined by its Fourier transform. ¤It is clear from Theorem 3.1 that sufficient conditions for (T (t), t ≥ 0) tobe G-covariant are[S(t), Ug] = 0 and µgt = µt, (3.5)for each t ≥ 0, g ∈ G. To establish whether these are also necessary, we mustprobe at the infinitesimal level and from now on, we make the additionalassumption thatUg(Dom(J)) ⊆ Dom(J),for all g ∈ G. Note that we can replace this by the weaker requirement thatJ has a Ug invariant core, for all g ∈ G, whenever such a core exists.Theorem 3.2 The following are equivalent:5(i) The Mehler semigroup is G-covariant.(iv) UgJx = JUgx and λg = λ,(v) [S(t), Ug] = 0 and µgt = µt,for all t ≥ 0, g ∈ G, x ∈ Dom(J).Proof. (i) ⇒ (iv) Apply (2.3) in Theorem 3.1 (iii) to obtain for eacht ≥ 0, g ∈ G, x, v ∈ H,exp{−∫ t0{λ(S(r)∗U−1g v)− λ(S(r)∗v)}dr}= exp{i〈v, [S(t), Ug]x〉}.Now take x ∈ Dom(J), differentiate both sides of the above equation and lett = 0 to obtainλ(v)− λg(v) = i〈v, [J, Ug]x〉.From this we deduce that the linear mapping from Dom(J) to H given byx → [J, Ug]x is constant. But 0 ∈ Dom(J) ⇒ [J, Ug]x = 0 for all x ∈Dom(J) ⇒ λg = λ.(iv) ⇒ (v) That UgJx = JUgx for all x ∈ Dom(J) ⇒ [S(t), Ug] = 0 is astandard (easily verified) fact about C0-semigroups. If λg = λ then by (2.3),we see thatµ̂tg(v) = exp{−∫ t0λ(S(r)∗U−1g v)dr}= exp{−∫ t0λ(U−1g S(r)∗v)dr}= exp{−∫ t0λg(S(r)∗v)dr}= µ̂t(v).But each µ̂tg = µ̂gt , hence µgt = µt, by the uniqueness of Fourier transforms.(v) ⇒ (i) is immediate ¤It follows from the Lévy-Khintchine formula (2.4) that λg = λ for allg ∈ G if and only ifUgb = b, Q = UgQU−1g , νg = ν, (3.6)for all g ∈ G. If (3.6) holds, we say that λ has G-invariant characteristics.6Corollary 3.1 If G acts irreducibly on H, then (T (t), t ≥ 0) is G-covariantif and only if1. There exists α ∈ R such that S(t) = eαtI for all t ≥ 0,2. λ has G-invariant characteristics with b = 0, Q = cI, for some c ≥ 0.In the case where H is infinite-dimensional, (2) is replaced by2′ λ has characteristics (0, 0, ν) where νg = ν, for all g ∈ G.Proof. The condition (1) on the semigroup is a consequence of Schur’slemma. Using (3.6), we see that for G-covariance, the ray {ρb, ρ ∈ R}must beinvariant under the action of G, hence b = 0 by irreducibility when dim(H) ≥2. When dim(H) = 1, b = 0 follows immediately from (3.6). From Schur’slemma again, we must have Q = cI with c ≥ 0, and in infinite dimensions Qcannot be trace class unless c = 0. ¤We recall that the irreducible representations of compact Lie groups arealways finite dimensional, so Corollary 3.1 makes most impact on noise re-duction when G is non-compact.4 Covariant Ornstein-Uhlenbeck ProcessesLet (Ω,F , (Ft, t ≥ 0), P ) be a stochastic base wherein the filtration (Ft, t ≥0) satisfies the usual hypotheses of completeness and right continuity and letX = (X(t), t ≥ 0) be a (time homogeneous) Markov process defined on Ωand taking values in H. For each t ≥ 0, x ∈ H, A ∈ B(H), pt(x,A) is thetransition probability P (X(t) ∈ A|X(0) = x). We say that X has covarianttransitions if for all t ≥ 0, x ∈ H,A ∈ B(H), g ∈ G:pt(Ugx, A) = pt(x, U−1g A). (4.7)Let (M(t), t ≥ 0) be the transition semigroup of X acting on Bb(H), so that(M(t)f)(x) = E(f(X(t))|X(0) = x) for each t ≥ 0, x ∈ H, f ∈ Bb(H).Proposition 4.1 (M(t), t ≥ 0) is G-covariant if and only if X has G-covariant transition probabilities.Proof. This follows from the facts that for each g ∈ G, x ∈ H, f ∈Bb(H), t ≥ 0,(UgM(t)f)(x) =∫Hf(y)pt(Ugx, dy), and(M(t)Ugf)(x) =∫Hf(Ugy)pt(x, dy) =∫Hf(y)pt(x, U−1g dy).7¤We recall that a Markov process X is G-invariant if it has invariant laws,i.e. for all t ≥ 0, A ∈ B(H), g ∈ G,pX(t)(A) = pX(t)(U−1g A).For the next result we assume that X is normal, i.e. the mapping x →pt(x,A) is measurable for each t ≥ 0, A ∈ B(H).Corollary 4.1 If the normal Markov process X has G-invariant transitionprobabilities and the law of X(0) is G-invariant, then X is G-invariant.Proof For each t ≥ 0, A ∈ B(H), g ∈ G,pX(t)(U−1g A) =∫HpX(0)(dx)pt(x, U−1g A)=∫HpX(0)(Ugdx)pt(Ugx,A)= pX(t)(A),where we have used the fact that each Ug is a bijection of H. ¤Now let Z = (Z(t), t ≥ 0) be a Lévy process, i.e. a càdlàg, Ft-adapted sto-chastically continuous process with stationary and independent incrementsfor which Z(0) = 0 (a.s.). Then there exists a negative-definite, hermitian,Sazonov continuous function λ : H → C with λ(0) = 0 such thatE(ei〈y,Z(t)〉) = e−tλ(y), (4.8)for each t ≥ 0, y ∈ H. Let (b,Q, ν) be the characteristics of λ.The Lévy-Itô decomposition within this context has been established byAlbeverio and Rüdiger ([1], see also [8]). It asserts that there exists a Brown-ian motion (BQ(t), t ≥ 0) with covariance operator Q and an independentPoisson random measure N on R+× (H −{0}) with intensity measure l⊗ ν(where l is Lebesgue measure on R+) such thatZ(t) = tb + BQ(t) +∫||x||<1xÑ(t, dx) +∫||x||≥1xN(t, dx), (4.9)where Ñ is the compensated Poisson measure, i.e. Ñ(dt, dx) = N(dt, dx)−dtν(dx).Now consider the stochastic differential equationdY (t) = JY (t)dt + dZ(t), (4.10)8with initial condition Y (0) = Y0, where Y0 is F0-measurable and independentof (Z(t), t ≥ 0). This has been given a precise meaning using stochasticintegration based on (4.9) in [2] and it is shown therein that it has a uniqueweak solution given by the Ornstein-Uhlenbeck processY (t) = S(t)Y (0) +∫ t0S(t− r)dZ(r), (4.11)for each t ≥ 0 (see [5] for an earlier alternative approach, and [4] for a methodfor obtaining strong solutions to (4.10) by extension to a larger Hilbert space).Y is a Markov process and its transition semigroup is a generalised Mehlersemigroup and so has the form (2.1).By Proposition 4.1, Y has covariant transition probabilities if and only if(T (t), t ≥ 0) is G-covariant. By Theorem 3.2, we see that this holds wheneverJ commutes with the group action and λg = λ, for all g ∈ G. By (4.8), thelatter holds if and only if Z is G-invariant. If G acts irreducibly on H, thenY is a classical Ornstein-Uhlenbeck process driven by G-invariant Z, so that(4.10) becomesdY (t) = αY (t)dt + dZ(t), (4.12)where α ∈ R.Example Let H = L2(Rd) and G be the orthogonal group O(d). Weconsider the representation of G on L2(Rd) given by (Lgf)(x) = f(g−1x), foreach g ∈ O(d), x ∈ Rd. Let (S(t), t ≥ 0) be the heat semigroup(S(t)f)(x) = (2πt)−d2∫Rdf(x + y)e−12t|y|2dy,for each t ≥ 0, f ∈ H, x ∈ Rd. It is well-known (and easily checked) that S(t)commutes with the group action. J is the usual Laplacian ∆ and its domainis the Sobolev spaceH2(Rd) ={f ∈ H;∫Rd|y|2|f̂(y)|2dy < ∞},where f̂ is the Fourier transform of f , which is easily seen to be invariantunder O(d). The group action is, of course, reducible and the most general(left) O(d)-covariant Ornstein-Uhlenbeck process is that driven by a (left)O(d)-invariant Lévy process satisfying the conditions (3.6), e.g. we may takedY (t) = ∆Y (t)dt + dBQ(t),where Q is any positive, self-adjoint trace class operator in the commutantof the real von Neumann algebra N = {Lg, g ∈ G}′′ (see Chapter 4 of [14]for a general account of such algebras). We may for example take Q = S(1).9If G is a non-compact group acting irreducibly in G, then Y is of the form(4.12) withZ(t) =∫||x||<1xÑ(t, dx) +∫||x||≥1xN(t, dx)for each t ≥ 0, and νg = ν for each g ∈ G. It is easily verified that a sufficientcondition for the latter is thatN(t, A)d= N(t, U−1g A),for each t ≥ 0, A ∈ B(H), g ∈ G. As an example, we take Z to be a G-invariant compound Poisson process,Z(t) = X1 + X2 + · · ·+ XN(t),where (Xn, n ∈ N) are i.i.d random variables with common G-invariant law qand (N(t), t ≥ 0) is an independent Poisson process with intensity h > 0. Inthis case ν(·) = hq(·) is a finite measure and we can write the unique weaksolution to (4.12) explicitly asY (t) = eαtY0 +∑n∈Neα(t−τn)Xn1[0,t](τn)for each t ≥ 0, where for each n ∈ N, τn is the nth arrival time for (N(t), t ≥ 0)and has a gamma distribution with density gn(x) =hn−1e−hxxn−1(n− 1)! 1(0,∞)(x).References[1] S.Albeverio, B.Rüdiger, Stochastic integrals and the Lévy-Itô decompo-sition on separable Banach spaces, Stoch. Anal. and Applns. 23, 217-53(2005)[2] D.Applebaum, Martingale-valued measures, Ornstein-Uhlenbeckprocesses with jumps and operator self-decomposability in Hilbertspace, to appear in Séminaire de Probabilités, 39.[3] O.E.Barndorff-Nielsen, N.Shephard, Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics,J.R.Statis.Soc. B, 63, 167-241 (2001)[4] V.I.Bogachev, M.Röckner, B.Schmuland, Generalised Mehler semi-groups and applications, Prob. Th. Rel. 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Covariant Mehler semigroups in Hilbert space

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Covariant Mehler semigroups in Hilbert space

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